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A014754 Primes p == 1 mod 8 such that 2 and -2 are both 4th powers (one implies other) mod p. 8
73, 89, 113, 233, 257, 281, 337, 353, 577, 593, 601, 617, 881, 937, 1033, 1049, 1097, 1153, 1193, 1201, 1217, 1249, 1289, 1433, 1481, 1553, 1601, 1609, 1721, 1753, 1777, 1801, 1889, 1913, 2089, 2113, 2129, 2273, 2281, 2393, 2441, 2473, 2593, 2657, 2689 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

Primes p such that x^4 == 2 has more than two (in fact four) solutions mod p. This is the sequence of terms common to A040098 (primes p such that x^4 == 2 has a solution mod p) and A007519 (primes of form 8n+1). Solutions mod p are represented by integers from 0 to p - 1. For p > 2, i is a solution mod p of x^4 == 2 iff p - i is a solution mod p of x^4 == 2, thus the sum of first and fourth solution is p and so is the sum of second and third solution. The solutions are given in A065909, A065910, A065911 and A065912. - Klaus Brockhaus, Nov 28 2001

Primes of the form x^2+64y^2. - T. D. Noe, May 13 2005

LINKS

N. J. A. Sloane and Vincenzo Librandi, Table of n, a(n) for n = 1..9769 (the first 1000 terms were found by Vincenzo Librandi)

PROG

(PARI) A014754(m) = local(p, s, x, z); forprime(p = 3, m, s = []; for(x = 0, p-1, if(x^4%p == 2%p, s = concat(s, [x]))); z = matsize(s)[2]; if(z>2, print1(p, ", ")))

(PARI) {a(n) = local(m, c, x); if( n<1, 0, c = 0; m = 1; while( c<n, m++; if( isprime(m), x = 0; for(y=1, sqrtint( m \ 64 ), if( issquare( m - 64 * y^2, &x), break)); if( x, c++ ))); m)} /* Michael Somos, Mar 22 2008 */

(PARI) forprime(p=1, 9999, p%8==1&&ispower(Mod(2, p), 4)&&print1(p", ")) \\ M. F. Hasler, Feb 18 2014

(PARI) is_A014754(p)={p%8==1&&ispower(Mod(2, p), 4)&&isprime(p)} \\ M. F. Hasler, Feb 18 2014

CROSSREFS

Cf. A040098, A007519, A014754, A007522, A065909, A065910, A065911, A065912, A070179.

Sequence in context: A039483 A104998 A033247 * A007766 A065111 A152308

Adjacent sequences:  A014751 A014752 A014753 * A014755 A014756 A014757

KEYWORD

nonn

AUTHOR

Warren D. Smith

EXTENSIONS

Removed erroneous Mma program; extended b-file using first PARI program of M. F. Hasler. - N. J. A. Sloane, Jun 06 2014

STATUS

approved

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Last modified February 18 00:19 EST 2019. Contains 320237 sequences. (Running on oeis4.)